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Results tagged with ap.analysis-of-pdes
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user 68232
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
5
votes
0
answers
147
views
weak maximum principle for weighted laplacian
Consider a radial weight $w=|x|^2 \geq 0$ for all $x\in \mathbb{R}^n$ and consider the operator
$$Lu= \frac{1}{w}\operatorname{div}(w\nabla u).$$
Then if $-Lu\leq 0$ on a smooth bounded domain $\Omega …
- 557
5
votes
1
answer
198
views
Convergence of heat flow on non-compact manifolds?
Consider the heat equation $\partial_t u= \Delta u+\lambda_1 u$ on a non-compact complete manifold $M$ (with nonpositive curvature) where $\lambda_1$ is the first eigenvalue and we start with some smo …
- 557
5
votes
1
answer
333
views
Does the Poincaré inequality hold on annular domains?
Does the following Poincaré inequality hold
$$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$
where $B_r$ denotes a ball of radius …
- 557
5
votes
2
answers
449
views
Question about Neumann eigenvalues on manifolds
Question:
Let $\Omega\subset \mathbb{S}^2_+$ be any geodesically convex subset of the hemisphere $\mathbb{S}^2_+$. Then is it true that $\mu_1(\Omega)\geq \mu_1(\mathbb{S}^2_+)$ where $\mu_1$ is the f …
- 557
4
votes
0
answers
171
views
Techniques for showing non-degeneracy results (PDE)
Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a …
- 557
4
votes
1
answer
330
views
Energy estimates for nonlinear wave type equation
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)$ …
- 557
4
votes
1
answer
135
views
Hölder Gradient Estimates for Linear Elliptic Equations in higher dimensions
I was looking at Theorem 12.4 of Gilbarg and Trudinger's Elliptic partial differential equations of second order (MR1814364, Zbl 1042.35002):
Theorem 12.4. Let $u$ be a bounded $C^2(\Omega)$ solution …
- 557
3
votes
0
answers
65
views
Existence of ground state solutions for the critical exponent
I have been recently reading Kwong's paper on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$.
The authors show that the above equation has a unique positive …
- 557
3
votes
1
answer
560
views
Does the following version of the Coifman–Meyer Theorem exist?
Recall the Coifman–Meyer Theorem as stated in Grafakos and Oh - The Kato-Ponce Inequality.
Theorem: Let $m \in L^{\infty}\left(\mathbf{R}^{2 n}\right)$ be smooth away from the origin and satisfy
$$
\l …
- 557
3
votes
0
answers
187
views
Parabolic smoothing for semilinear PDE
Consider the semilinear energy-critical parabolic PDE in $\mathbb{R}^3$
\begin{align}
\partial_t u &= \Delta u + |u|^{4/(n-2)}u = \Delta u + u^5\\
u(0,x) &= u_0\in \smash{\dot{H}}^1(\mathbb{R}^n).
\en …
- 557
2
votes
1
answer
202
views
Kernel for an equation involving the Ornstein-Uhlenbeck operator
Consider the following PDE on $\Omega\subset \mathbb{R}^n$ for $n\geq 2:$
\begin{align}
\Delta u - x\cdot \nabla u &= f(x),\text{ in } \Omega\\
u&=0 \text{ on }\partial \Omega
\end{align}
Are there an …
- 557
2
votes
0
answers
183
views
Are solutions to this elliptic PDE uniformly bounded in $\mathbb{R}^n?$
Given a fixed value $\lambda>0$ let $u\in \dot{H}^1(\mathbb{R}^n)$ be a weak solution (or eigenfunction with eigenvalue $\lambda$) in dimension $n\geq 3$ to the following PDE,
$$-\Delta u = \lambda \r …
- 557
2
votes
0
answers
136
views
Computing the fractional laplacian of a logarithm function
Are there explicit formulas to compute the fractional laplacian $(-\Delta)^{s/2}\log |x|$ in $\mathbb{R}^n$ for $s\in (0,2)$?
- 557
2
votes
1
answer
99
views
How to estimate higher order regularity for wave type equation with time dependant coefficie...
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)$ …
- 557
2
votes
0
answers
505
views
Interpolation inequality involving negative Sobolev space
$\newcommand\norm[1]{\left\|#1\right\|}\newcommand\inner[2]{\langle #1,#2\rangle}$
Let $u\in \dot{H}^1(\mathbb{R}^n)$ for $n\geq 3$ where $\dot{H}^{1}$ denotes the homogeneous Sobolev space that is th …
- 557